5



5. Quantum mechanics in one dimension

Schrödinger’s equation is the analogue to the wave equation of sound, light, water,[pic], which works for all classical waves, that have either a photon associated with it or a pseudo-particle (such as a phonon)

harmonic wave, plane wave is solution to this equation moving to the right

y(x,t) = y0 cos 2 π (x/λ - t/T) = y0 cos 2π/λ (x – vt) where v = λ/T

these functions describe something with physical significance, e.g. the E vector, the amplitude on a water wave, the air pressure in a sound wave

the square of that function y(x,t)2 ~ η energy per unit volume, intensity (I) of waves is energy density times wave speed, so I ~ y(x,t)2 ~ η can be put down to number of photons (or pseudo-particles) at anyone place at a certain time, is also the likelihood of finding a photon (or pseudo-particle) there (x) and then (t)

what we need is something altogether different but mathematically similar - a wave equation for matter wave, the solutions to which, i.e. Ψ(x,t) - the matter waves - will be a valid description of how “small” things move – and the square of which Ψ(x,t)2 will give us the probability of finding the particle of the matter wave there (x) and then (t)

Ψ(x,t) contains everything that is and can be know about the particle, to get the probability of finding the particle at some specific (x,t) we have to calculate Ψ(x,t)2 Born’s interpretation

call the probability that particle will be found in the infinitesimal small interval dx about the point x P(x), probability density as it is per length unit, then Born’s interpretation is

P(x) dx = Ψ(x,t)2 dx will be a number ≤ 1 (at time t)

it is not possible to specify with certainty the position of a particle (x) - Heisenberg’s uncertainty principle – but it is possible to assign definitive values of probabilities for observing it at any place we care to calculate the square function for (at a given time)

Ψ(x,t)2 is intensity of matter wave, a measurable quantity, while Ψ(x,t) is only a mathematical model for the matter wave, a non physical thing, can’t be measured

Schrödinger’s equation equivalent to Newton’s second law, (Solutions to Newton’s second law described how things move at the macroscopic scale!!! Newton’ second law contained the solution of Newton’s first law, Schrödinger equation will contain equivalent to Newton’s firs law a free particle, plane wave, harmonic wave and superpositions of plane waves describing a pulse)

Schrödinger developed his equation after his prior attempts to explain with de Broglie’s relation the Bohr model at a more fundamental level failed, a colleague told him one does need a wave equation to make progress with waves, so Schrödinger boned up on the maths and found the one that works for all matter waves !!!

Partial derivates and complex numbers

suppose we have a function f(x,y) of two variables and want to know how this function varies with one variable only, say x

we treat the other variable y as a constant and differentiate f(x,y) with respect to x

result is called a partial derivate and written as

[pic]

rules of ordinary differentiation apply

e.g. f = f(x,y) = yx2 [pic]as y is a constant

on the other hand[pic]as x is now a constant

second order partial derivates [pic]are calculated by repeating the procedure

e.g. f = f(x,y) = yx2 [pic] as y is again a constant

application on something more challenging

classical wave equation is [pic]

for electromagnetic wave, sound wave, standing wave on a guitar, water wave, wave on a very long string free to travel

solutions of classical wave equation for monochromatic (ω = constant) undamped (A = constant) wave traveling the right is

y(x,t) = Ae-iω(t-x/v)

now show that y = Ae-iω(t-x/v) is a solution to the classical wave equation

first partial derivate of y with respect to x

[pic]

second derivate [pic]

first partial derivate with respect to t

[pic]

second derivate [pic]

comparing the second derivates, difference is just [pic]otherwise they are identical so

[pic]which is the wave equation,

so y(x,t) = Ae-iω(t-x/v) must be a solution to this equation

complex wave functions / just like complex numbers

Ψ = A + iB, A real part of function

B imaginary part

then Ψ* = A – iB ,

(i is replace everywhere by – i and one has the conjugate complex function)

Ψ2 = Ψ* Ψ = Ψ Ψ* = A2 – i2 B2 = A2 + B2 is all real

i2 = -1

the fundamental problem of quantum mechanics

given the wave function at some instant, say t = 0, i.e. Ψ(x,0), find the wave function at some or all other times t - when there are forces acting on the particle

Ψ(x,0) is the initial information on the particle,

Newton’s mechanics analogue was initial position (x) and momentum (p) of a classical particle,

now it is an infinite set of numbers a set of values, for all points x one value of Ψ(x,0)

in Newton’s mechanics we obtain x(t) and p(t) by solving Newton’s second law [pic], an net force acting on the particle changed it’s momentum, change in position over kinematics

Schrödinger’s equation (SE) propagates Ψ(x,0) forward in time,

that’s what we want to know, given (within Heisenberg’s uncertainty) we know where a particle is and what its momentum there is, we want to calculate were will be at some time (t) and what will it’s momentum be at that time

i.e. the initial Ψ(x,0) changes into Ψ(x,t)

[pic]

F = [pic] is the force acting on the particle

U(x) is the potential energy function of the Force

1. left hand side (LHS) of SE is first evaluated for Ψ(x,0), i.e. t = 0, as it is not dependent on time, i.e. we make partial derivations and add the influence of the potential energy function on Ψ(x,0)

LHS of SE equals right hand side (RHS) of SE result must be equal to [pic]at t = 0, i.e. initial rate of change of wave function

2. from [pic]at t = 0, RHS of SE, we compute Ψ(x,dt), the wave function at an infitesimal small time interval (δt) later by superposition

Ψ(x,δt) = Ψ(x,0) + [[pic]]0 δt

3. that results gets plucked in at LHS of SE again, but now we evaluate Ψ(x,δt), i.e. this time make the partial derivations for Ψ(x,δt) add the influence of the potential energy function on Ψ(x,δt) (just like we did for t = 0, first step), result is again equal to RHS of SE

4. from [pic]at t = δt, RHS of SE, we compute Ψ(x,dt2), the wave function at an infitesimal small time interval (δt2) later by superposition

Ψ(x,δt2) = Ψ(x, δt) + [[pic]]δt δt2

……

each such repetition advances Ψ (x,δtn-1) one step in time δtn forward

until we have the time (t > 0) we want to investigate our particle again – it can all be done by computer quickly and numerically

---------------

“Somebody could still asks: How does it work? What mechanism is represented by the wave function? Nobody has ever found a mechanism behind the wave function. Nobody can explain more that we have just discussed. Nobody will give you an explanation about what is going on at a deeper level. As a matter of fact, we do not have an inkling about a basic mechanism from which the wave function could be derived.” R. P. Feynman, 1971

numerical solutions of Schrödinger equations are fine but how may one obtain a mathematical expression for Ψ(x,t)

mathematical procedure called separation of variables,

Ψ(x,t) = [pic]

if U(x) potential energy is function of x only (not of t) !!!

[pic]

[pic]

with E = h f = 2π [pic]f = ω [pic] so ω = [pic]

we can look at the e-iωt factor above which describes the time dependency if the potential energy does not depend on time – so that time dependency factor is [pic]

in equations above, E is the total energy, which we can normalize to be the kinetic energy plus the potential energy, (if we set rest energy E0= 0, as a reference form which energy is counted - which we can do arbitrarily)

[pic]

rearranged for further use and called, time independent, steady-state, or stationary Schrödinger equation in one dimensions

if we have an arbitrary potential energy function U(x) there are no explicit analytical solutions to this equation

[pic] must be “well behaved” just as Ψ has to in order to give sensible results for probabilities, i.e. finite everywhere including +- [pic], single valued for any x, continuous,

and “smooth” – which is [pic]must also be continuous and single valued (the Serway book says here: wherever U(x) has a finite value, other books say all the time)

– all of them are mathematical conditions, so called boundary conditions

if we can separate the variables,

we also get Ψ(x,t)2 = [pic], meaning all probabilities we calculate from Ψ(x,t) will not depend on time, are static or stationary

expansion to three dimensions straightforward

[pic]

consequence at least 3 quantum numbers, taking account of the spin of the electron it will be 4 for electrons confined to be in an atom

let’s look at a free particle in the plane wave approximation, also called a harmonic wave

free non-relativistic particle means no force on it F = 0 = [pic], no force means no potential energy U(x), and no dependence of the potential energy on t, as particle is free, all energy is kinetic E = ½m v2

one dimensional time independent SE simplifies to

[pic]

½m v2 can be rewritten as [pic], multiplying within the straight bracket yields [[pic]]

p = h/λ and [pic] so [pic] = (2π/λ )2 = k2 per definition of wave number

[pic]

[pic]

[pic]

[pic]

[pic]

[pic] most general

are all solution of one dimensional time independent Schrödinger equation, where A and B are arbitrary constants (such constants appear generally in solutions to the SE and we will define then in the normalization process)

we had

Ψ(x,t) = [pic]

so in order to get most general solution of time dependent SE

Ψ(x,t) we multiply most general time independent solution [pic]with time dependence [pic]

Ψ(x,t) = [pic]= [pic]= [pic]

where is that free particle? answer: calculate Ψ(x,t)2

remember any function (be it exponential or sinusoidal) of from (kx ± ωt) represents a traveling wave

for (kx - ωt) wave is traveling to the right

for (kx +ωt) wave is traveling to the left,

lets decide our particle should travel to the right, we can do that by setting B = 0 in the most general solution

so Ψ(x,t)2= Ψ* Ψ = [pic]

so the probability is a constant A2 = Ψ0(x,t)2 at all places and times

we may have as well calculated [pic]to find the probability of finding the particle for any x we want

[pic]= [pic][pic] = [pic]

[pic]

analyzing the graph we see that the probability of finding the particle in any one segment of equal length Δx or dx is absolutely the same as it is a constant, so the particle has equal probabilities to be at any place, there is no most likely place

so let’s assume we have a free particle moving to the right, expressed by wave function, see what happens if we put it into Schrödinger equation

Ψ(x,t) = [pic]

where A is a constant, let’s differentiate partially for x and t and put our derivates into the time dependent (one dimensional) Schrödinger equation

[pic]

[pic]

as it is a free particle, it is not under the influence of a force, so it has constant (time and position independent) net potential energy U(x) = U0, which may be zero or any other value (remember potential energy levels can be set arbitrarily)

[pic]

plugging our derivates in

[pic] which we can divide by Ψ!!!

and we get

[pic]

as we know [pic]

so what is [pic] with k2 = (2π / λ)2 and p2 = (h / λ)2 = m2v2

[pic]= ½ m v2 is kinetic energy of the free particle moving

to the right QED, formalism makes sense

as long as there is no net force, a particle does not change momentum, and moves in a straight line at constant speed, uniform linear motion – just the same for macroscopic particles is stated in Newton’s first law,

Newton’s first law is contained in, i.e. it is actually a solution of Newton’s second law, just as harmonic (plane) wave is a solution of, i.e. is (contained in), Schrödinger’s law

free particle solution can also be written as

Ψ(x,t) = [pic]

with E = h f = 2π [pic]f and λ = h /p = [pic]

Ψ(x,t) =

[pic]where E = [pic]= KE + PE

Let’s look again at probability density, normalization and boundary conditions

normalization:

P(x) dx = Ψ(x,t)2 dx

is probability that particle will be found in infinitesimal interval dx about the point x,

P(x) is called probability density (here in m-1also

m-2 or m-3

as probability has to be a single value at every (x) point we care to look at to make sense, Ψ(x,t) and Ψ(x,t)2 have to be single valued and continuous functions (of x and t) to make sense, in addition, they have to be smooth

general solutions to the Schrödinger equation contain arbitrary constants which we can arbitrarily assign values to, so a good idea is to use these constants for normalization procedures

If we know the particle must be somewhere (within some length, or area, or volume for which we have precise values, e.g. x1 and x2 (x1 < x2 ) or even infinite values +-[pic]we specify the arbitrary constant so that

[pic] meaning the particle does exist between x1 and x2 with 100 % certainty at all times

any wave function which satisfied this conditions is said to be normalized Ψ(x,t)

if we have such a normalized Ψ(x,t), we can calculate the probability of the particles existence between a and b, where a ≥ x1 and b ≤ x2 in % by

[pic] so if we forget to normalize

we have just P ~ probability of finding the particle there and then, with normalization this becomes a measure in %

in all cases P is just the area under a curve

[pic]

this sets a strict condition to Ψ if it is not only to be a function that happens to solve the Schrödinger equation, but also to represent the pilot/guiding/matter wave of a real particle

the area under the curve has to be finite so that it can normalized to be 1 or 100 %, so Ψ(x,t) has to go to zero for x1 and x2 otherwise Ψ(x,t)2 would not go to zero and the area under the curve would not be finite

example: Bohr radius in hydrogen atom

[pic]

boundary conditions must be fulfilled for Ψ(x,t) to represent a real particle

well behaved functions

Ψ and [pic] must be “well behaved” in order to give sensible results for probabilities,

i.e. finite everywhere,

single valued for any x (and t),

continuous, i.e. having x (and t) values everywhere (unless V(x) is infinite)

and “smooth” – which is [pic]must also be continuous wherever U(x) has a finite value

in addition to being a solution of the Schrödinger equation

so boundary conditions and requirements of normalization will make it possible for us to decide which solution of Schrödinger equation represent real particles and which are a purely mathematical construct

example free “particle”

note that the solution of the

Schrödinger equation that

describes this particle can

not be (easily) normalized, as

the area under the parallel

line reaching from – to +

infinity is infinite!!!, that,

however, was implied by the

definition above

so this wave function does not describe a “real physical”

particle, it is however a very useful starting model for a real particle as we can construct form such waves by means of superposition an acceptable model for a real particle, just as we did for electromagnetic waves in chapter 4

that real particle will then have a wave function that can be normalized and the plot of it probability density function will show a pulse with has finite values at some region [pic], peaking somewhere, and going to zero everywhere else, especially when x approaches +- infinity

let’s look at the uncertainty principle again,

[pic]

and apply it to the free particle in the graph above

if the particle is free, no net force acts on it, Newton’s 1st law states, if there is no net force acting, there is no change in momentum, so

[pic] which amounts to a violation of the uncertainty principle, a model can violate the principle, but not a real particle, so the free particle described by one plane wave function (rather that a sum of many plane wave functions) is not a real particle

having a free particle described by a pulse will again mean we have mathematical uncertainties

Δx Δk ≈ 1

Δω Δt ≈ 1

in the model that translate to real physical uncertainties when we make a physical interpretation of the model by replacing Δk with [pic](after de Broglie) and multiplying both sides with [pic]

(analogously: applying the definition of ω = 2π f and E = h f (Plank-Einstein equation) gives physical meaning to Δω Δt ≈ 1)

so we don’t violate with the mathematical model for the pulse/wave bundle/wave packet Heisenberg’s uncertainty principle and this describes a real particle,

in addition, the area under a pulse will of course be finite, so we can normalize our wave function

Expectation values and Operators

the solutions to the Schrödinger equation contain everything that can be known (i.e. which the uncertainty principle allows us to know) about the movement of an entity that is a wave-particle with mass

so lets extract the (arithmetic) mean position – which is also called the expectation value, (your book states here incorrectly the average position, an average does not refer to a distribution/population but the arithmetic mean does),

[pic]

where f is the dimensionless frequency of occurrence of one particular value of x

(forget about p 215 lower half and p 216 top paragraph, I am pretty sure that is incorrect as I did not find a similar Modern Physics treatment in Beiser and Tipler

short maths into

if the “sample of x values” is large the mean of these values may be taken as an estimate of the distribution/population mean

the sum of all discrepancies form the mean is zero

[pic]

the variance of the mean

[pic]

for large n, one can approximate n with n-1 and use the variance of the population

[pic]as a measure of variance of

the sample

standard deviation (σ) is the square root of the variance and another measure of the amount of scatter in the data

if σ = 0 then var(x) = 0, there is no spread in the data and the distribution is called sharp

the uncertainly principle now tells us that particle positions (x) can only been know with probabilities, i.e. its distribution is never sharp and always fuzzy

back to the expectation value,

[pic] where Ψ(x,t) has to be

normalized

definition

the arithmetic mean of x that would be expected from measurements of the positions of a large number of particles with the same wave function!

don’t confuse with probability of finding a particle in an infinitesimal interval around x – it’s completely different things, so P = 0 may be compatible with a finite expectation value

e.g. for an infinite square well and even quantum number wave functions: P(L/2) = 0, but = L/2 because Ψ2 and also[pic]2 are symmetric about that point

to calculate we have the definition of the expectation value

[pic]

we need normalized wave functions, and they are

[pic]

as there is no i(s) [pic], the conjugate complex function has the same form and the [pic]2 are simply

[pic]

so the integral becomes

[pic]

[pic]

since sin (nπ) = 0, cos (2nπ) = 1 and cos 0 = 1, for all values of n the expectation value of x is

[pic]

in all quantum states, the arithmetic mean position of the particle is in the middle of the box

for n = 2,4,6 the “average” position is also L/2 and this has nothing to do with [pic]the probability density of finding the particle there

[pic]

now the expectation value of any function of x can be calculated the same way

[pic]

so f(x) can be potential energy U(x) for example

however no function p = p(x) exist by virtue of the uncertainty principle, Δpx Δx ≥[pic] if both of these entities vary in a fuzzy way there simply can’t be a relation between the two of them (there is simply no classical path in quantum mechanics)

p = mv but p ≠ p(x) in quantum mechanics

there is the same problem with expectation value of E, a there is an uncertainty principle as well ΔE Δt ≥[pic] only if we are considering a stationary state, i.e. when there is no time dependency and no Δt, no such uncertainty, we will have sharp values for energy

so what we need here are operators

operator is a mathematical concept telling us what to do with the operand that follows it

e.g. [pic] (x2 t) means that one has to take the partial x derivate of the function (x2 t) and multiply it with [pic]

so [pic] (x2 t) = [pic]

what is [pic] (cos x ) = -[pic] sin x

entities for which we have operators are called observables as they have physical meaning and can be observed (although subject to the uncertainty principle)

now [pic] is actually the momentum operator [p] that gives us the expectation value of the momentum

[pic] note that the order of factors is important, there is only one way of doing it correctly

similarly

[pic]

first one operator is applied to its operand yielding the operand for the second operator (which will again stand to the right of the operator)

for example: calculate the expectation value for the ground state wave function in the infinite square well,

we know it is a stationary state (standing wave) so it is time independent, we know the particle is trapped in the well, so it is never outside, so we can restrict the integral to the well

the (normalized and time independent) wave function for that state is [pic] [pic] as there are no i(s) in it the conjugate complex of that functions [pic] is also [pic]

so [pic]

simplifies to [pic]

this is of course because sin x = 0 at the nodes!!

so the expectation value is zero, what does it mean, simply the particle is just as likely moving to the right as it is moving to the left, the arithmetic mean must, thus, give zero

generally operators are written in sharp straight brackets, i.e. [p] or with a “caret”, i.e. [pic]

as there are many more observables, entities with physical meaning that are allowed to be known by the uncertainty principle, there are many more operators that give us expectation values of these observables

total energy operator [E] = [pic]

kinetic energy operator non relativistic

[KE] = [pic]

Potential energy operator [(PE)] = [U] = U(x)

now let’s see if everything is consistent with the Schrödinger equation

E = KE + U so we must also have [E] = [KE] +[U]

that is equivalent to [pic]

now we multiply both sides with [pic](it has to come from the left as these “guys” are operators)

and get [pic]

so postulating both

[E] = [pic]

[p]=[pic]

is equivalent to postulating the Schrödinger equation !!!

now notice the operators of kinetic and potential energy are only involving spatial coordinate x, we can define a combined operator for the total energy that also involves only the spatial coordinate x, this is call the Hamiltonian operator [H]

[H] = [pic]

sum of kinetic and potential energy operator must also be total energy operator that involves only time coordinate (t)

E = [pic]

so we have actually two total energy operators and if they operate on the same wave function, the must yield the same observable expectation value !! again multiplying with [pic]yields

[H][pic]= [E][pic]

the “pretty compact” version of the Schrödinger equation

Eigenvalues and Eigenfunctions

for simplicity we deal here only with time independent wave functions, if something is in a steady state the uncertainty principle ΔE Δt ≥ [pic] does not apply, there is all the time in the world, so the energy has settled into a stationary state an exact value, it is only when it jumps between stationary states that there is a Δt again, and with it an uncertainty of energy that shows up in a widths of a spectral line

“eigen” is German and means self, so what is meant here is combinations of real numbers (values) and functions that are equivalent to the action of an operator on these functions.

(if you know about systems of linear equations and matrix representation, you have the very same things, combinations of vectors with values that are “self” solutions to the problem, was invented in Göttingen by Jordan and Hilbert, who told Born und Heisenberg about it, …)

mathematical definition [G]Ψn = gnΨn where

e.g. operator [pic] has eigen function [pic]

what is the eigenvalue to this functions and operator

[pic]

as the eigen function was just [pic] the (generally real) number 4 is for that function exactly equivalent to the operator [pic]

back to physics

eigenfuctions are here again solutions to the Schrödinger equation, we deal only with time independent form if we are looking at stationary states such as in the case of a particle in a box we get sharp values for certain operators such as the total energy operator, so there is no expectation value for energy as there is no arithmetic mean of measurements on many identical particles, if we are dealing with an eigenvalue/eigenvector problem the eigenvalue is just one value, e.g. a definitive energy for every eigenfunction, , eigenfunction and eigenvectors are refereeing to a set of quantum numbers that are integers

from particle in an infinite square well, you know, energy comes only in discrete values, En, these are the eigenvalues to the eigenfunctions [pic]

so time independent Schrödinger equation can be written most compactly

[H][pic]= En[pic]

for correct description of atoms we will have a second set of eigenvalues and eigenfuctions, because angular momentum is in nature also quantized not only energy, so there will be another quantum number actually there will be two more sets of eigenfunctions and eigenvalues as a state of an electron in an atom is described by 4 quantum numbers

Model: Particle in a box with infinitely large potential barriers, infinite square well

infinite barriers, the particle is always confined, never outside

exercise: deriving form of the wave function under the boundary conditions

it’s a stationary state, so we use time independent Schrödinger equation

[pic]

so [pic] outside the box

inside box U(x) = 0

[pic]

solutions of this ordinary partial equation are sin kx and cos kx

so most general solution is

[pic] inside the box 0 < x 0

but if total E is smaller than the height of the potential energy walls of the well, i.e. E-PE = E-U < 0, there is no kinetic energy left to roam freely, so classical particle can’t be outside the wall and moving, it is trapped forever in 0 < x < L

In quantum mechanics, because of the condition

[pic] or [pic] must be continuous, i.e. slopes must be continuous,

a particle leaks out into the potential walls !!!

This is because Ψ is never zero outside the well, so the probability of finding the particle there Ψ2 is not zero either, so the particle is actually there !!!

so lets look at the parts of the wave function that penetrated into potential walls

solutions to the (time independent) Schrödinger equation

[pic] for x < 0, section I, where C is a constant we can

use for fit to the second segment

[pic] for x > L, section III, where D is a constant we

can use for fit to the second segment

and [pic] is a positive constant, as U is taken to be zero in the well and some positive value outside the well

so we have an exponential decay, that’s pretty fast,

from of the constant α, we can see that the heavier the particle is and/or the larger the difference U – E, (i.e. the larger – KE of the bound state) the faster [pic]decays in the walls, if the walls are infinitely wide, the wave function decays to zero

general solution for region II is

[pic] as U(x) = 0 in

the well

and k = [pic] as usual

but as sin (0) = 0 we have to set A = 0 and can only use the second part with “cos” functions

that “cos” function’ has to match with the functions for section I and III at x = 0 and L and its first derivate with respect to x has to match as well (smoothness condition of wave functions that describes real particles) for x = 0 and L

this can only be achieved for certain energy levels En which are all smaller that their counterparts from the infinite square well of the same widths

[pic]

example say we have n = 1 and [pic], i.e. [pic]at the wall is only half the maximal value of this function at the center

so we can say cos kL = 1/2

kL = 60° = π/3

k = π/3L k = [pic] from above, resolved for E1

[pic]

for that particular scenario exactly 2.25 times smaller due to the particular height and widths of the square potential well, i.e. U and L, that results in the value of the wave function at L just being half the maximum value (which we have in the center of the well)

on can also see form the graph that the wavelength that fit into a finite square well (with leakage into the barriers) are somewhat larger than those wavelength that would fit into an infinite square well of the same widths, larger wavelength correspond after de Broglie (λ = h / p) to smaller momenta (p), and momenta are liked to kinetic energy by KE = [pic]

the similarity between finite and infinite square well is also expressed in the concept of a penetration depth δ

per definition [pic]

at a distance δ beyond each of the well edges the amplitude of the wave function has fallen to 1/e of its value at the edges, and approached zero exponentially, i.e. very very fast beyond δ

with that we can make an approximation

[pic] (which shall be sufficiently

accurate if δ En

as wave function for a particle that can make a transition, we need time dependent wave function Ψ(x,t), as it is two different states m and n, we have a superposition

Ψm,n(x,t) = a Ψm(x,t) + b Ψn(x,t)

initially say a = 1, b = 0, electron in excited state, m

while in transition a < 1, b ................
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